Extension of a theorem of Shi and Tam
Extension of a theorem of Shi and Tam
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DOI：
10.1007/s00526-011-0402-2
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发表时间：
2009-11
影响因子：
2.1
通讯作者：
M. Eichmair;P. Miao;Xiaodong Wang
中科院分区：
文献类型：
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作者：
M. Eichmair;P. Miao;Xiaodong Wang
本文证明了Shi和Tam （J Differ Geom 62:79-125, 2002）的一个定理的以下推广：设（Ω，g）为n≥3维紧致黎曼流形，自旋于bbb7，具有非负标量曲率和平均凸边界。如果每个边界分量Σihas正标量曲率和等距嵌入为平均凸星形超曲面，则其中his的平均曲率Σiin （Ω，g）是in的欧几里得平均曲率，其中σ和表示各自的体积形式。此外，对于某些边界分量Σiif，且仅当（Ω，g）与中的定义域等距时，等式成立。在证明中，我们使用了Gerhardt （J Differ Geom 32:299 - 314,1990）和Urbas （Math Z 205(3): 355-372, 1990）研究的流内的外部叶状。我们还仔细地建立了低维的刚性陈述，没有使用Shi和Tam中使用的自旋假设（J Differ Geom 62:79-125, 2002）。
In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω,g) be ann-dimensional (n≥ 3) compact Riemannian manifold, spin whenn> 7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σihas positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface, thenwhereHis the mean curvature of Σiin (Ω,g),is the Euclidean mean curvature ofin, and wheredσ anddenote the respective volume forms. Moreover, equality holds for some boundary component Σiif, and only if, (Ω,g) is isometric to a domain in. In the proof, we make use of a foliation of the exterior of the’s inby the-flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).